Grade 6 - Mathematics1.43 Relation Between G.C.D. and L.C.M.

The greatest common divisor (GCF) of two positive whole numbers is the product of the prime factors common to both numbers.

The least common multiple (LCM) is the smallest multiple common to both numbers.

Sometimes these terms are written as greatest common divisor GCD, and lowest common denominator LCD, when they are being used with fractions.

Relationship between G.C.D and L.C.M. through an example:Example: Consider the numbers 8 and 12.
Let us find the L.C.M. of 8 and 12.
8 = 2 x 2 x 2
12 = 2 x 2 x 3
Collect the largest number of occurrence of each prime number.
The largest number of times the factor 2 occurs is three.
The largest number of times the factor 3 occurs is one.
Therefore the L.C.M. is = 2 x 2 x 2 x 3 = 24.

Now let us find the G.C.D. of 8 and 12.
8 = 2 x 2 x 2
12 = 2 x 2 x 3
We observe that 2, 2 are the common prime factors of 8 and 12.
Hence the G.C.D. of 8 and 12 is = 2 x 2 = 4.

Now, we find the product of G.C.D. and L.C.M.,
that is LCM x GCD = 24 x 4 = 96.

Now we shall find the product of two given numbers.
i.e., 1st number x 2nd number = 8 x 12 = 96.

From the above example, we find that 1st number x 2nd number = LCM x GCD.

Hence, the product of two numbers is equal to the product of LCM and GCD of the two numbers.